Who do you prefer, the Spice Girls or All Saints? All Saints would contend that there's no comparison. Whenever the media set up a rivalry between two such groups, whether it's the Beatles versus the Stones or Take That versus East 17, the latter band will always object that their competitors are not competitors at all, but exist in a different sphere. One group makes children's pop music; the other is more mature, more authentic and - although this is usually left unsaid - more black. In practice, this tends to mean that they don't smile much and they don't have as many tunes.

Mercifully, All Saints didn't try to be too adult at Wednesday's concert. Since "Never Ever" made a home in the top 10, the group's follow-up singles have spent less and less time in the charts, while the Saints themselves have spent more and more time in the tabloids, either falling pregnant, falling out with each other or falling into the arms of Robbie Williams (Nicole), Jamie Theakston (Natalie), Damon Albarn (Shaznay) and some bloke from Jamiroquai (Mel). This tour, their first, is All Saints' opportunity to remind us that they're pop stars. And given that the bar's takings won't match those of the lollipop sellers, the way to do this is to put on a kids' show - and no stinting on the dance routines and costumes. (Note to all choreographers and designers: the trilby gimmick is old hat. Kitting out your subjects as gangsters has been done far too many times since Michael Jackson's "Smooth Criminal".)

All Saints are determined to retain some edge though, and they carefully balance their street cred against their Sesame Street cred. The stage is strewn with packing crates for that warehouse-party vibe, and the musicians are black and include an authentically mountainous guitarist wearing a baseball cap, and a dreadlocked DJ who inquires repeatedly whether Nuttin'ham is in the house. On the other hand, the Saints' girlie banter is aimed straight at Smash Hits' younger readers: when Shaznay introduces the band, she asks us whether we fancy the keyboard players (she doesn't ask about the mountainous guitarist). There is something here for schoolgirls and adults alike. And the teenage boys in front of me got to grin nervously at the sight of Natalie's bra.

You could quibble about the egregious sound quality or the feeble singing: it's never an encouraging omen when a group comprising four female vocalists employs two female backing vocalists. But on the whole, All Saints stood up well. Their music is sophisticated and sassy; unlike the Spice Girls' records, All Saints' album can be enjoyed by people whose age is greater than their shoe size.

And there was one other aspect that was fundamental to the concert's success. The women looked so much cheerier on stage than they ever did on television. Mind you, having to get through that terrible spoken bit at the start of "Never Ever" for the thousandth time would make anyone miserable.

Halfway through his show, Keigo Oyamada, aka Cornelius, played a note- perfect "Love Me Tender" on a theremin, while on a screen behind him, Elvis Presley played it silently on an acoustic guitar. It was a profoundly resonant, symbolic tableau: the man from the land of karaoke, adored in Britain like no Japanese pop star before him, was appropriating a Western icon while using a device which is still most strongly associated with the Beach Boys' "Good Vibrations". No? Well, it was a fun moment, anyway.

Cornelius - named after a noble chimp in Planet of the Apes - has often been called a Beach Boys/ Beastie Boys hybrid, and the description is almost as accurate as it is catchy, although to do him justice you'd have to add Bach and Blur to the gene-pool. Last year's album, Fantasma, is incredible: a dizzying collage of live drums and jungle beats; acoustic guitars and synthesisers; Californian harmonies and samples; mellow lounge music and manic cartoon soundtracks. For years we've been shipping off our worst pop stars to make a living in Japan; now Japan has sent their best one back here.

With two new remix albums to promote, Cornelius performed on Tuesday in the Scala, a swanky converted cinema that has just reopened near King's Cross station. Maybe it was being in London that affected him, but he left most of the Bach/ Beach/Beastie influence at home, and concentrated on Blur-ry, spikey punk rock, battered out by a band in matching jeans and combat shirts. To begin with, it was disappointing to hear the album roughed up like this, but it was soon entertaining in its own right. Oyamada is an unashamed guitar hero, as cool as it is possible to be with a bowl cut. And his band are a thrilling collision between splashy-drummed garage-rock chaos and being very tight, very organised and, as someone near me muttered, "very Japanese".

Monday's show by Tom Petty and the Heartbreakers was their first in the UK for seven years, but Petty must have seemed a familiar figure to all cartoon aficionados. The way he mumbles between songs in an impenetrable Southern accent ... that pointed nose and those shadow-casting cheekbones ... if he cut his wispy blond hair, he'd be a dead ringer for Boomhauer from King of the Hill.

Hank Hill and his beer buddies would approve of Petty. He has been influenced by contemporary trends over the decades, but he and the Heartbreakers are essentially custodians of good old-fashioned, down-the-line American music. They are a safe pair of hands: rock'n'roll curators. The first song of the evening was Chuck Berry's "Around and Around", and this only bolstered the impression that Petty is a keeper of the flame rather than someone with incendiary talent of his own.

None the less, his albums have been of consistent quality since 1976, right up to the new one, Echo. And you have to warm to someone who loves music as openly as Petty does. "There's no good reason to play this song," he grinned as he introduced "Guitar Boogie Shuffle" (which was exactly that), "except that I really do like it." The only other bit of Boomhauerese I could decipher was Petty's opinion that writing a song is like having an orgasm: he doesn't understand the mechanism, he's just glad it happens.

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